32 Logic as a Tool (g) (p → r) ∧ (q → r) and (p ∧ q ) → r (h) (p → r) ∨ (q → r) and (p∨ q ) → r (i) ((p∧q )→ r) and (p → r )∨(q → r) (j) p → (q → r) and (p → q ) → r (k) p ↔ (q ↔ r) and q ↔ (p ↔ r ) (l) p →(q → r) and (p → q ) → (p → r ) 1.3.4 Negate each of the following propositional formulae and transform the result to an equivalent formula in a negation normal form (a) (p ∨ ¬q ) ∧ ¬p (b) (p → q ) ∨ (¬p → ¬q ) (c) (p → ¬q ) → p (d) p → (¬q → p) (e) (p ↔ ¬q ) → ¬r (f) p → (¬q ↔ r ) Augustus De Morgan (27.6.1806–18.3.1871) was a British mathematician, logician and a popularizer of mathematics Influenced by George Boole, he pioneered the application of algebraic methods to the study of logic in the mid-19th century, becoming one of the founding fathers of modern mathematical logic In particular, he was the first to formulate the logical equivalences now known as de Morgan’s laws De Morgan was born in Madura, India and became blind in one eye soon after his birth He graduated from Trinity